Perturbations of completely positive maps and strong NF algebras

TL;DR

This paper investigates conditions under which completely positive maps can be approximated by complete order embeddings, and applies these results to classify certain $C^*$-algebras as strong NF algebras.

Contribution

It establishes new criteria for perturbing completely positive maps to embeddings and applies this to characterize a class of $C^*$-algebras as strong NF algebras.

Findings

Perturbation of completely positive maps to embeddings under specific conditions.

Existence of examples where such perturbations are impossible.

Identification of $C^*$-algebras with $ ext{OL}_ ext{infty}=1$ as strong NF algebras.

Abstract

Let $\phi:M_n\to B(H)$ be an injective, completely positive contraction with $\V\phi^{-1}:\phi(M_n)\to M_n\V_{cb}\leq1+\delta(\epsilon).$ We show that if either (i) $\phi(M_n)$ is faithful modulo the compact operators or (ii) $\phi(M_n)$ approximately contains a rank 1 projection, then there is a complete order embedding $\psi:M_n\to B(H)$ with $\V\phi-\psi\V_{cb}<\epsilon.$ We also give examples showing that such a perturbation does not exist in general. As an application, we show that every $C^*$-algebra $A$ with $\mathcal{OL}_\infty(A)=1$ and a finite separating family of primitive ideals is a strong NF algebra, providing a partial answer to a question of Junge, Ozawa and Ruan.